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Fast-growing hierarchy
A fast-growing hierarchy (FGH) is a certain hierarchy mapping ordinals \(\alpha < \mu\) to functions \(f_\alpha: \mathbb{N} \rightarrow \mathbb{N}\). For large ordinals \(\alpha\), \(f_\alpha\) grows very rapidly. Due to its simple and clear definition, as well as its origins in professional mathematics, FGH is a popular benchmark for large number functions. Definition A fast-growing hierarchy consists of an ordinal \(\mu\) and a fundamental sequence system \(S: \mu \cap \text{Lim} \rightarrow (\mathbb{Z}_0 \rightarrow \mu)\), where \(S(\alpha)(n)\) is denoted \(\alphan\). The semantics are as follows: * \(f_0(n) = n + 1\) * \(f_{\alpha+1}(n) = f^n_\alpha(n)\), where \(f^n\) denotes function iteration * \(f_\alpha(n) = f_{\alphan}(n)\) if and only if \(\alpha\) is a limit ordinal The general case where \(f_0\) is any increasing function forms a fast iteration hierarchy. For a fast-growing hierarchy to be useful to googologists, it is also expected to satisfy the property that for all \(\alpha < \beta < \mu\), \(f_\alpha\) is eventually dominated by \(f_\beta\). Wainer hierarchy Definitions of the fast-growing hierarchy and choices of fundamental sequence systems vary between authors, so it is generally problematic to speak of "the" fast-growing hierarchy. The most well-known FGH, however, is the Wainer hierarchy, which has \(\mu = \varepsilon_0 + 1\) and an FS system defined as follows: *\(\omegan = n\) *\(\omega^{\alpha + 1}n = \omega^\alpha n\) *\(\omega^{\alpha}n = \omega^{\alphan}\) if and only if \(\alpha\) is a limit ordinal *\((\omega^{\alpha_1} + \omega^{\alpha_2} + \cdots + \omega^{\alpha_{k - 1}} + \omega^{\alpha_k})n = \omega^{\alpha_1} + \omega^{\alpha_2} + \cdots + \omega^{\alpha_{k - 1}} + \omega^{\alpha_k}n\) where \(\alpha_1 \geq \alpha_2 \geq \cdots \geq \alpha_{k - 1} \geq \alpha_k\) *\(\epsilon_00 = 0\) (alternatively \(1\)) and \(\epsilon_0+ 1 = \omega^{\epsilon_0n}\) For example, the fundamental sequence for \(\omega^\omega\) is \(1, \omega, \omega^2, \omega^3, \ldots\). When authors refer to "the fast-growing hierarchy" without clarification, the Wainer hierarchy is usually meant. Approximations Below are some functions in the Wainer hierarchy compared to other googological notations. There are a few things to note: *Relationships denoted \(f_\alpha(n) > g(n)\) hold for sufficiently large \(n\), not necessarily all \(n\). *\(m\) indicates any positive integer. *\(^ab\) indicates tetration. *\(\uparrow\) indicates arrow notation. *\(\text{Ack}\) indicates Harvey Friedman's single-argument Ackermann function \(\text{Ack}(n)\). *\(\lbrace \rbrace\) indicates BEAF. *Expressions with \(X\) indicate structures. \begin{eqnarray*} f_0(n) &=& n + 1 \\ f_1(n) &=& f_0^n(n) = ( \cdots ((n + 1) + 1) + \cdots + 1) = n + n = 2n \\ f_2(n) &=& f_1^n(n) = 2(2(\ldots 2(2n))) = 2^n n > 2 \uparrow n \\ f_3(n) &>& 2\uparrow\uparrow n \\ f_4(n) &>& 2\uparrow\uparrow\uparrow n \\ f_m(n) &>& 2\uparrow^{m-1} n \\ f_\omega(n) &>& 2\uparrow^{n-1} n = Ack(n) \\ f_{\omega+1}(n) &>& \lbrace n,n,1,2 \rbrace \\ f_{\omega+2}(n) &>& \lbrace n,n,2,2 \rbrace \\ f_{\omega+m}(n) &>& \lbrace n,n,m,2 \rbrace \\ f_{\omega2}(n) &>& \lbrace n,n,n,2 \rbrace \\ f_{\omega3}(n) &>& \lbrace n,n,n,3 \rbrace \\ f_{\omega m}(n) &>& \lbrace n,n,n,m \rbrace \\ f_{\omega^2}(n) &>& \lbrace n,n,n,n \rbrace \\ f_{\omega^3}(n) &>& \lbrace n,n,n,n,n \rbrace \\ f_{\omega^m}(n) &>& \lbrace n,m+2 (1) 2 \rbrace \\ f_{\omega^{\omega}}(n) &>& \lbrace n,n+2 (1) 2 \rbrace > \lbrace n,n (1) 2 \rbrace \\ f_{\omega^{\omega}+1}(n) &>& \lbrace n,n,2 (1) 2 \rbrace \\ f_{\omega^{\omega}+2}(n) &>& \lbrace n,n,3 (1) 2 \rbrace \\ f_{\omega^{\omega}+m}(n) &>& \lbrace n,n,m+1 (1) 2 \rbrace \\ f_{\omega^{\omega}+\omega}(n) &>& \lbrace n,n,n+1 (1) 2 \rbrace > \lbrace n,n,n (1) 2 \rbrace \\ f_{\omega^{\omega}+\omega+1}(n) &>& \lbrace n,n,1,2 (1) 2 \rbrace \\ f_{\omega^{\omega}+\omega2}(n) &>& \lbrace n,n,n,2 (1) 2 \rbrace \\ f_{\omega^{\omega}+\omega^2}(n) &>& \lbrace n,n,n,n (1) 2 \rbrace \\ f_ 2}(n) &>& \lbrace n,n (1) 3 \rbrace \\ f_ 3}(n) &>& \lbrace n,n (1) 4 \rbrace \\ f_ m}(n) &>& \lbrace n,n (1) m+1 \rbrace \\ f_{\omega^{\omega+1}}(n) &>& \lbrace n,n (1) n+1 \rbrace > \lbrace n,n (1) n \rbrace \\ f_{\omega^{\omega+2}}(n) &>& \lbrace n,n (1) n,n \rbrace \\ f_{\omega^{\omega+3}}(n) &>& \lbrace n,n,n (1) n,n,n \rbrace \\ f_{\omega^{\omega+m}}(n) &>& \lbrace n,m (1)(1) 2 \rbrace \\ f_{\omega^{\omega2}}(n) &>& \lbrace n,n (1)(1) 2 \rbrace = \lbrace n,2 (2) 2 \rbrace \\ f_{\omega^{\omega3}}(n) &>& \lbrace n,n (1)(1)(1) 2 \rbrace = \lbrace n,3 (2) 2 \rbrace \\ f_{\omega^{\omega m}}(n) &>& \lbrace n,m (2) 2 \rbrace \\ f_{\omega^{\omega^2}}(n) &>& \lbrace n,n (2) 2 \rbrace \\ f_{\omega^{\omega^3}}(n) &>& \lbrace n,n (3) 2 \rbrace \\ f_{\omega^{\omega^m}}(n) &>& \lbrace n,n (m) 2 \rbrace \\ f_{\omega^{\omega^\omega}}(n) &>& \lbrace n,n (n) 2 \rbrace = \lbrace n,n (0,1) 2 \rbrace \\ f_{^4{\omega}}(n) &>& \lbrace n,n ((1) 1) 2 \rbrace = n \uparrow\uparrow 3 \&\ n \\ f_{^5{\omega}}(n) &>& \lbrace n,n ((0,1) 1) 2 \rbrace = n \uparrow\uparrow 4 \&\ n \\ f_{^6{\omega}}(n) &>& \lbrace n,n (((1) 1) 1) 2 \rbrace = n \uparrow\uparrow 5 \&\ n \\ f_{^m{\omega}}(n) &>& X \uparrow\uparrow m-1 \&\ n \\ f_{\varepsilon_0}(n) &>& X \uparrow\uparrow n-1 \&\ n \end{eqnarray*} Buchholz hierarchy In [http://epub.ub.uni-muenchen.de/3842/1/buchholz_wilfried_3842.pdf An Independence Result for (\(\Pi_1^1\)-CA) + BI], Wilfried Buchholz discusses an ordinal hierarchy where \(\mu = \psi_0\varepsilon_{\Omega_\omega + 1}\), where \(\psi\) is Buchholz's ordinal collapsing function and \(\psi_0\varepsilon_{\Omega_\omega + 1}\) is the TFB ordinal. Other hierarchies We can easily replace the Wainer hierarchy with a different one, such as a hierarchy that works for much larger ordinals such as the small Veblen ordinal or the Takeuti-Feferman-Buchholz ordinal. As long as fundamental sequences are defined, it is possible to extend FGH up to an arbitrary countable ordinal. However, it is impossible to create an effective system that works for all countable ordinals, since a system of fundamental sequences up to \(\omega_1\) would be nonconstructive. Selecting fundamental sequences is not an easy problem, since some selections can lead to pathological hierarchies where \(\alpha < \beta\) does not necessarily imply \(f_\alpha <^* f_\beta\) (where \(<^*\) means eventual domination). In a 1976 paper, Diana SchmidtSchmidt, Diana. Built-Up Systems of Fundamental Sequences and Hierarchies of Number-Theoretical Functions proved a theorem that is useful for identifying and guarding against pathological hierarchies. Given a fundamental sequence system, define \(P(\beta + 1) = \beta\) and \(\forall \beta \in \text{Lim} : P(\beta) = \beta0\). A fundamental sequence system is built-up if, for all \(\alpha\) and \(n\), \(\alphan = P^k(\alphan+1)\) for some \(1 \leq k < \omega\). Schmidt showed that a built-up fundamental sequence system guarantees that all \(f_\alpha\) are monotonically increasing and that \(\alpha < \beta \Rightarrow f_\alpha <^* f_\beta\). It is possible to define the fast-growing hierarchy for all recursive ordinals, and even for nonrecursive countable ordinals. However, the definitions will necessarily be nonrecursive, making analysis far more complicated. To our knowledge, there has been no research into the creation of nonrecursive fast-growing hierarchies. See also *List of googological functions *Slow-growing hierarchy *Hardy hierarchy *Middle-Growing Hierarchy *Wow function Sources ja:急増加関数 Category:Hierarchies Category:Functions